Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that

$$|f|_{var}= \sum_{n=1}^{\infty} var_n f < \infty,$$

where $var_n f = sup\{|f(x)-f(y)|: x,y \in X , x_k = y_k, \forall k = 1,\dots, n\}$. Also, suppose that $\sum_{i =1}^\infty \exp(\sup_{x \in [i]}f(x)) < \infty$, where $[i] = \{x \in X: x_1 = i\}$. We want to conclude that for $t>1$ exists a constant for $C_t$ such that

$$\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp( (t-1) sup_{x \in [i]}f(x)) < C_t.$$

To be more precise, I am trying to understand the proof of the Lemma 3.1 of the I. Morrison's following paper: "Entropy for zero-temperature limits of Gibbs-equilibrium states for countable-alphabet subshifts of finite type". But I think that understand this less general case, I understand the paper by myself. Thank you very much.